| L(s) = 1 | − 2.69·2-s + 1.88·3-s + 5.28·4-s − 2.05·5-s − 5.08·6-s + 0.784·7-s − 8.85·8-s + 0.544·9-s + 5.54·10-s − 2.34·11-s + 9.94·12-s + 4.30·13-s − 2.11·14-s − 3.86·15-s + 13.3·16-s − 2.48·17-s − 1.46·18-s + 19-s − 10.8·20-s + 1.47·21-s + 6.33·22-s + 4.84·23-s − 16.6·24-s − 0.775·25-s − 11.6·26-s − 4.62·27-s + 4.14·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 1.08·3-s + 2.64·4-s − 0.919·5-s − 2.07·6-s + 0.296·7-s − 3.13·8-s + 0.181·9-s + 1.75·10-s − 0.707·11-s + 2.87·12-s + 1.19·13-s − 0.565·14-s − 0.999·15-s + 3.33·16-s − 0.602·17-s − 0.346·18-s + 0.229·19-s − 2.42·20-s + 0.322·21-s + 1.35·22-s + 1.01·23-s − 3.40·24-s − 0.155·25-s − 2.28·26-s − 0.889·27-s + 0.783·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 - 1.88T + 3T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 7 | \( 1 - 0.784T + 7T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 23 | \( 1 - 4.84T + 23T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 - 0.527T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 + 4.10T + 43T^{2} \) |
| 47 | \( 1 - 0.0151T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 5.59T + 61T^{2} \) |
| 67 | \( 1 - 5.88T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 1.58T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 1.28T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.189394315664239352008668145329, −7.84074563550135001122636384570, −7.04988911434076315146843708005, −6.36010392570464692534834943512, −5.19795649564204076711582798047, −3.77713470944741547006481726853, −3.09485909857551803467195516364, −2.27306290898595368362006480308, −1.27956800908664683595804817989, 0,
1.27956800908664683595804817989, 2.27306290898595368362006480308, 3.09485909857551803467195516364, 3.77713470944741547006481726853, 5.19795649564204076711582798047, 6.36010392570464692534834943512, 7.04988911434076315146843708005, 7.84074563550135001122636384570, 8.189394315664239352008668145329
